A plane flies 300 kilometer on a bearing 30 degrees. It then flies 450 kilometer on a bearing 270 degrees. (a) Find the bearing and distance of the plane from its starting point. (b) Find how far north and how far west the plane from its starting point. Number two, the angle of elevation of the top of a radio mast from a point 54 meter from its base on level ground is 60 degrees. Find the height of the mast to the nearest meter. Use scale, one centimeter represents 9 meters.

For a bearing θ (clockwise from North):
North component = Distance × cos(θ)
East component = Distance × sin(θ)

Leg 1: 300 km on a bearing of 30°
North₁ = 300 km × cos(30°) = 300 km × (√3 / 2) = 150√3 km
East₁ = 300 km × sin(30°) = 300 km × (1/2) = 150 km

Leg 2: 450 km on a bearing of 270° (due West)
North₂ = 450 km × cos(270°) = 450 km × 0 = 0 km
East₂ = 450 km × sin(270°) = 450 km × (-1) = -450 km (or 450 km West)

Total North displacement (N_total) = North₁ + North₂
N_total = 150√3 km + 0 km = 150√3 km ≈ 259.81 km

Total East displacement (E_total) = East₁ + East₂
E_total = 150 km + (-450 km) = -300 km (or 300 km West)

The total distance (D) is the hypotenuse of a right triangle formed by N_total and E_total.
D = √(N_total² + E_total²)
D = √((150√3)² + (-300)²)
D = √( (22500 × 3) + 90000 )
D = √( 67500 + 90000 )
D = √(157500)
D = 150√7 km
D ≈ 396.9 km (to one decimal place)

The plane is North (N_total is positive) and West (E_total is negative) of the starting point.
Let α be the angle measured from the North axis towards the West.
tan(α) = |E_total| / |N_total|
tan(α) = 300 / (150√3) = 2 / √3
α = arctan(2/√3) ≈ 49.1°

The bearing is measured clockwise from North. Since the displacement is in the North-West quadrant, the bearing is 360° - α.
Bearing = 360° - 49.1° = 310.9°

We have a right-angled triangle where:

• The angle of elevation is 60°.
• The distance from the base (adjacent side) is 54 m.
• The height of the mast (opposite side) is what we need to find. The tangent function relates the opposite and adjacent sides: tan(angle) = Opposite / Adjacent.

tan(60°) = Height / 54 m

Height = 54 m × tan(60°)
Height = 54 m × √3
Height ≈ 54 m × 1.73205
Height ≈ 93.5287 m

Height ≈ 94 m